Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > df-har | Structured version Visualization version GIF version |
Description: Define the Hartogs
function , which maps all sets to the smallest
ordinal that cannot be injected into the given set. In the important
special case where 𝑥 is an ordinal, this is the
cardinal successor
operation.
Traditionally, the Hartogs number of a set is written ℵ(𝑋) and the cardinal successor 𝑋 +; we use functional notation for this, and cannot use the aleph symbol because it is taken for the enumerating function of the infinite initial ordinals df-aleph 9369. Some authors define the Hartogs number of a set to be the least *infinite* ordinal which does not inject into it, thus causing the range to consist only of alephs. We use the simpler definition where the value can be any successor cardinal. (Contributed by Stefan O'Rear, 11-Feb-2015.) |
Ref | Expression |
---|---|
df-har | ⊢ har = (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦 ≼ 𝑥}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | char 9020 | . 2 class har | |
2 | vx | . . 3 setvar 𝑥 | |
3 | cvv 3494 | . . 3 class V | |
4 | vy | . . . . . 6 setvar 𝑦 | |
5 | 4 | cv 1536 | . . . . 5 class 𝑦 |
6 | 2 | cv 1536 | . . . . 5 class 𝑥 |
7 | cdom 8507 | . . . . 5 class ≼ | |
8 | 5, 6, 7 | wbr 5066 | . . . 4 wff 𝑦 ≼ 𝑥 |
9 | con0 6191 | . . . 4 class On | |
10 | 8, 4, 9 | crab 3142 | . . 3 class {𝑦 ∈ On ∣ 𝑦 ≼ 𝑥} |
11 | 2, 3, 10 | cmpt 5146 | . 2 class (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦 ≼ 𝑥}) |
12 | 1, 11 | wceq 1537 | 1 wff har = (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦 ≼ 𝑥}) |
Colors of variables: wff setvar class |
This definition is referenced by: harf 9024 harval 9026 |
Copyright terms: Public domain | W3C validator |